25 OPENING THE BLACK BOX: STRUCTURAL FACTOR MODELS …

Econometric Theory, 25, 2009, 1319–1347. Printed in the United States of America.doi:10.1017/S026646660809052X

OPENING THE BLACK BOX:STRUCTURAL FACTOR MODELSWITH LARGE CROSS SECTIONS

MARIO FORNIUniversita di Modena e Reggio Emilia

andCEPR

DOMENICO GIANNONEECARES, Universite Libre de Bruxelles

MARCO LIPPIUniversita di Roma “La Sapienza”

LUCREZIA REICHLINEuropean Central Bank

ECARES, Universite Libre de BruxellesandCEPR

This paper shows how large-dimensional dynamic factor models are suitable forstructural analysis. We argue that all identification schemes employed in structuralvector autoregression (SVAR) analysis can be easily adapted in dynamic factor mod-els. Moreover, the “problem of fundamentalness,” which is intractable in SVARs,can be solved, provided that the impulse-response functions are sufficiently hetero-geneous. We provide consistent estimators for the impulse-response functions andfor (n,T ) rates of convergence. An exercise with U.S. macroeconomic data showsthat our solution of the fundamentalness problem may have important empirical con-sequences.

1. INTRODUCTION

Recent literature has shown that large-dimensional approximate (or generalized)dynamic factor models can be used successfully to forecast macroeconomic vari-ables (Forni, Hallin, Lippi, and Reichlin, 2005; Stock and Watson, 2002a, 2002b;Boivin and Ng, 2003; Giannone, Reichlin, and Sala, 2005). These models assume

We thank, for suggestions and criticism, Manfred Deistler, Marc Hallin, two anonymous referees, and Pentti Saikko-nen, co-editor of Econometric Theory. M. Forni and M. Lippi are grateful to the Italian Ministry of University andResearch for financial support. D. Giannone and L. Reichlin were supported by a PAI contract of the Belgian federalgovernment and an ARC grant of the Communaute Francaise de Belgique. Address correspondence to Marco Lippi,Dipartimento di Economia, Via Cesalpino 12, I-00161 Roma, Italy; e-mail: [email protected].

c© 2009 Cambridge University Press 0266-4666/09 $15.00 1319

1320 MARIO FORNI ET AL.

that each time series in the data set can be expressed as the sum of two orthogonalcomponents: the “common component,” capturing that part of the series that co-moves with the rest of the economy, and the “idiosyncratic component,” whichis the residual. The vector of the common components is highly singular, i.e., isdriven by a very small number (as compared to the number of variables) of shocks(the “common shocks” or “common factors”). Indeed, evidence based on differentdata sets points to the robust finding that few shocks explain the bulk of dynamicsof macro data (see Sargent and Sims, 1977; Giannone, Reichlin, and Sala, 2002;Giannone et al., 2005). If the common component of the variable to be predictedis large, a forecasting method based on a projection on linear combinations ofthese shocks performs well because, although being parsimonious, it captures therelevant comovements in the economy.

Here we argue that the scope of dynamic factor models goes beyond forecast-ing. Our aim is to open the black box of these models and show how statisticalconstructs such as factors can be related to macroeconomic shocks and their prop-agation mechanisms.

We define macroeconomic shocks as those structural sources of variation thatare cross-sectionally pervasive, i.e., that significantly affect most of the variablesof the economy, as opposed to idiosyncratic sources of variation that are specificto a single variable or a small group of variables, hence capturing both sectoral-local dynamics (let us say “micro” dynamics) and measurement error. Our aim isidentification of the macroeconomic shocks and their dynamic effect on macro-economic variables, whereas the idiosyncratic components are disregarded.

A key paper in which the distinction between macroeconomic shocks andidiosyncratic sources of variation is systematically exploited for macroeconomicmodeling is Sargent and Sims (1977), in which several models, both “Keynes-ian” and “classical,” are reformulated as factor models with a small number ofmacroeconomic shocks. More recent literature includes papers in which dynamicstochastic general equilibria (DSGE), augmented with measurement errors, areestimated by maximum likelihood (augmenting a theory-based model with mea-surement errors goes back to Sargent, 1989; see also Altug, 1989, and the litera-ture mentioned therein; Ireland, 2004, and the literature mentioned therein; for anexplicit link to factor models see Giannone, Reichlin, and Sala, 2006; Boivin andGiannoni, 2006).

The approach we propose here is a combination of structural vector autoregres-sion (SVAR) analysis and large-dimensional dynamic factor models. Precisely,the factor model is used to consistently estimate common and idiosyncratic com-ponents of macroeconomic variables. Then we apply SVAR analysis to identifythe relationship between common components and macroeconomic shocks.

Our approach differs from error-augmented DSGE models in that we estimatethe impulse-response functions of the macroeconomic variables to macroeco-nomic shocks without imposing any theory-based dynamic restriction. It has aclose relationship to factor augmented autoregression (FAVAR) models, in whicha vector autoregression (VAR) is augmented with common factors (see Bernanke,

OPENING THE BLACK BOX 1321

Boivin, and Eliasz, 2005). The link between factor models, FAVAR models, andVAR models has been studied by Stock and Watson (2005), who show how SVARtechniques can be used in a factor-model context. However, our analysis of thefundamentalness of the structural shocks in factor models, and the consequentmotivation for an autoregressive approximation (see the discussion that followsand Section 3), is a distinctive feature of the present paper. An early work inwhich a large factor model is used for structural analysis is Forni and Reichlin(1998); major differences with the present paper are the empirical focus and theproposed estimation procedure.

To give a brief outline of the structure of the paper, suppose that we are inter-ested in key macroeconomic variables such as per capita consumption, income,and investment, denoted by ct , yt , and it (see our empirical exercise in Section 5).The macrovariables ct , yt , and it are embedded in a large macroeconomic dataset (the number of variables in our exercise is 89) and modeled as a commoncomponent, driven by structural macroeconomic shocks, plus an idiosyncraticcomponent (variable specific shocks and measurement error). Under fairly gen-eral assumptions the common components can be estimated consistently (seeSection 2).

The vector of the common components, call it χχχnt , has dimension n, the num-ber of variables in the data set, and rank q, the number of macroeconomic shocks(three in our exercise), and is therefore highly singular. A crucial step in our analy-sis is the dynamic specification of χχχnt as a (singular) vector autoregression drivenby the macroeconomic shocks. This implies assuming that the macroeconomicshocks are fundamental for the common components χχχnt . Section 3 is dedicatedto showing that the fundamentalness problem, a weakness of SVAR analysis,finds a satisfactory solution within our approach (on the fundamentalness issuein SVAR models see Hansen and Sargent, 1991; Lippi and Reichlin, 1993, 1994;and, more recently, Chari, Kehoe, and McGrattan, 2005; Fernandez-Villaverde,Rubio-Ramirez, and Sargent, 2005; Giannone et al., 2006). Nonfundamentalnessof structural shocks is a consequence—this is the usual explanation—of the agentshaving an information set that is larger than the econometrician’s. We argue thatin large-dimensional factor models, in which the number of observed variables islarger than the number of shocks (unlike in SVAR models), such “superior infor-mation” can occur only by a fluke (on the importance of this feature for monetarymodels, see Bernanke and Boivin, 2003; Giannone et al., 2002, 2005).

Once the vector autoregressive specification for χχχnt has been motivated, weshow that all the identification techniques developed in SVAR analysis, such aslong-run or impact effects, can be successfully imported in the identification ofstructural macroeconomic shocks within large-dimensional dynamic factor mod-els. As in SVAR analysis, the structural shocks are obtained by linearly transform-ing the estimated residual vector vvv t , the key difference being that here the numberof shocks q is smaller than the number of variables. Last, we can go back tothe variables of interest and study their dynamic response to structural macroeco-nomic shocks. Section 5 analyzes an empirical example on U.S. macroeconomic


data that revisits the results of King, Plosser, Stock, and Watson (1991) in the lightof our discussion on fundamentalness.

Section 4 studies consistency and rates of convergence for the estimators of theshocks and the impulse-response functions.

2. THE LARGE-DIMENSIONAL DYNAMIC FACTOR MODEL

The dynamic factor model used in this paper is a special case of the generalizeddynamic factor model of Forni, Hallin, Lippi, and Reichlin (2000) and Forni andLippi (2001). Such a model, and the one used here, differs from the traditionaldynamic factor model of Sargent and Sims (1977) and Geweke (1977), in that thenumber of cross-sectional variables is infinite and the idiosyncratic componentsare allowed to be mutually correlated to some extent, along the lines of Cham-berlain (1983), Chamberlain and Rothschild (1983), and Connor and Korajczyk(1988). Closely related models have been recently studied by Stock and Watson(2002a, 2002b), Bai and Ng (2002), and Bai (2003).

Denote by xxxTn = (xit )i=1,...,n; t=1,...,T an n × T rectangular array of observa-

tions.

Assumption 1. xxxTn is a finite realization of a real-valued stochastic process

XXX = {xit , i ∈ N, t ∈ Z , xit ∈ L2(�,F, P)}indexed by N×Z, where the n-dimensional vector processes

{xxxnt = (x1t . . . xnt )′, t ∈ Z}, n ∈ N,

are stationary, with zero mean and finite second-order moments �xk =E[xnt x′

n,t−k],k ∈ Z.

We assume that each variable xit is the sum of two unobservable compo-nents, the common component χi t and the idiosyncratic component ξi t . The com-mon components are driven by q common shocks uuut = (u1t u2t . . . uqt )

′. Note thatq is independent of n (and small as compared to n in empirical applications).Precisely, defining χχχnt = (χ1t . . . χnt )

′ and ξξξnt = (ξ1t . . . ξnt )′:

xxxnt =χχχnt +ξξξnt ,

χχχnt = Bn(L)uuut ,(1)

where the following conditions hold.

Assumption 2. uuut is a q-dimensional orthonormal white noise, and Bn(L) isa nested sequence of one-sided n × q absolutely summable matrix polynomials(infinite in general). Moreover, there exist an integer r ≥ q, a nested sequence ofn ×r matrices An , and a one-sided absolutely summable r ×q matrix polynomial(infinite in general) N (L), such that

Bn(L) = An N (L). (2)


Defining the r ×1 vector fff t as

fff t = N (L)uuut , (3)

(1) can be rewritten in the static form

xxxnt = An fff t +ξξξnt . (4)

In what follows, we shall use the term static factors to denote the r entries offff t , whereas the common shocks uuut will be also referred to as dynamic factors.

The dynamic factors uuut and Bn(L) are assumed to be structural sources of vari-ation and impulse-response functions, respectively. Therefore model (1), as spec-ified in Assumptions 1 and 2 and the other assumptions that follow, is a structuralfactor model.

Obviously xxxnt admits infinitely many different representations of the forms (1)and (4), with different dynamic and static factors. In particular, if H is an orthog-onal q × q matrix, then χχχnt = Cn(L)vvv t , with vvv t = Huuut , Cn(L) = Bn(L)H ′ (thesame applies to the static factors with H replaced by any invertible r ×r matrix).In Sections 3 and 4.1 we discuss identification of uuut , i.e., the conditions underwhich Bn(L) and uuut can be determined among all alternative impulse-responsefunctions and dynamic factors.

Assumption 3 (Orthogonality of common and idiosyncratic components). Forall n, the vector ξξξnt is stationary. Moreover, uuut is orthogonal to ξiτ , i ∈N, t ∈Z,τ ∈ Z.

The assumption of orthogonality between common and idiosyncratic compo-nents has an economic justification. Interpreting the factor model as the jointmodel of the economy and the statistical agency, under reasonable hypotheseson the behavior of the statistical agency, the latter is orthogonal to the signal cap-tured, in our framework, by the common shocks (for a discussion, see Sargent,1989). Moreover, orthogonality between common and idiosyncratic componentsensures that the entries of Bn(L) can be interpreted as impulse-response functionsof the common shocks on the χ ’s and on the variables xit themselves.

Some definitions are needed for the next two assumptions. Let �χk be the

k-lag covariance matrix of χχχnt and denote by μχj the j th eigenvalue, in decreas-

ing order, of �χ0 . Moreover, let �χ(θ) and �ξ(θ) be the spectral density matrix

of χχχnt and ξξξnt , respectively, and denote by λχj (θ) and λ

ξj (θ) their eigenvalues as

functions of θ ∈ [−π π ], in decreasing order.To avoid heavy notation, indication of the dependence on n and T is kept to

a minimum. In particular, dependence on n of �χk , μ

χj , etc., just defined, and of

other scalars and matrices defined subsequently, is not made explicit. In the sameway, reference to T and n will be avoided for estimated scalars and matrices. Forexample, the estimator of �x

0 , the covariance matrix of xxxnt , is denoted by �x0 .


Assumption 4 (Pervasiveness of common dynamic and static factors).

(a) As n → ∞ we have λχq (θ) → ∞ for θ almost everywhere (a.e.) in [−π π ].

(b) There exist constants cj ,cj , j = 1, . . . ,r , such that cj > cj+1, j = 1, . . . ,r −1, and

0 < cj < liminfn→∞ n−1μ

χj ≤ limsup

n→∞n−1μ

χj ≤ cj .

Assumption 5 (Nonpervasiveness of the idiosyncratic components). There ex-ists a realL such that λξ

1(θ) ≤L for any n ∈N and θ a.e. in [−π π ]. This obviously

implies that μξ1 ≤ L for any n ∈ N, μ

ξj being the j th eigenvalue of �

ξ0 .

Assumption 5 includes the case in which the idiosyncratic components are mu-tually orthogonal with an upper bound for the spectral densities (and therefore forthe variances). Mutual orthogonality is the usual condition in finite-dimensionalfactor models. Assumption 3 relaxes such condition by allowing for a limitedamount of cross correlation among the idiosyncratic components. Assumption 4(pervasiveness of the common factors) implies that each of the common shocksujt affects (almost) all the variables xit , i ∈ N, with nondeclining coefficients.

Some comments on our assumptions are in order.

1. Assumption 4(a) implies that the number q of dynamic factors and the com-mon components χi t are unique; i.e., a representation of the form (1)–(4)with a different number of dynamic factors or different common componentsis not possible (see Forni and Lippi, 2001).

2. Assumption 4(b) implies that the number r of static factors is unique; i.e., astatic representation of the common components χi t with a different numberof static factors is not possible.

3. We define the static and dynamic rank of fff t as the rank of, respectively,its variance-covariance and spectral density matrix. By Assumption 4(a) thedynamic rank of fff t is q for θ a.e. in [−π π ]. Assumption 4(b) entails that,for n sufficiently large, An has full rank r and that fff t has static rank r for anygiven t . Thus, for any given t , the space spanned by χi t , i ∈N, coincides withthe space spanned by the static factors f j t , j = 1, . . . ,r , and has thereforedimension r .

The following dynamic factor model has been often considered in the large-dimensional factor-model literature (see Stock and Watson, 2002a, 2002b, 2005;Bai and Ng, 2007; Forni et al., 2005):

χχχnt = Cn0 fff ∗t +Cn1 fff ∗

t−1 +·· ·+Cns fff ∗t−s, (5)

where fff ∗t is q-dimensional and the matrices C are n × q and nested and fff ∗

t hasthe VAR representation

�(L) fff ∗t = (1−�1L −·· ·−�h Lh) fff ∗

t = uuut , (6)


where �(L) is q ×q. Using the definitions

fff t =(

fff ∗′t fff ∗′

t−1 . . . fff ∗′t−s

)′, An = (Cn0 Cn1 . . .Cns),

N (L) = (K (L)K (L)L . . . K (L)Ls)′,

where K (L) = (�(L)′)−1, we have fff t = N (L)uuut and

xxxnt = An fff t +ξξξnt . (7)

The static rank of fff t is always q(s +1). However, for (7) to be a static represen-tation of the model it is necessary that An be full rank, and this depends on thecoefficients of the matrices Cnj :

1. If no restrictions among the coefficients of the matrices Cnj hold (assume,e.g., that they are independently drawn from the same distribution), then (7)is a static representation of the model.

2. If restrictions hold such that An is not full rank, then r < q(s + 1), and ob-taining a static representation requires further manipulation. For example,assume that q = 1, s = 1, so that (5) can be written as χi t = ci0ut +ci1ut−1.If no restrictions hold among the c’s, then r = 2, and (7) is a static represen-tation. But if the restriction ci1 = aci0 holds, then r = 1, N (L) = 1 + aL ,ft = (1+aL)ut , and An = (c10c20 . . . cn0)

′.

In any case, with or without restrictions, existence of a static representation formodel (5)–(6) is an immediate consequence of the following remark.

Remark R. Assume that χχχnt = Bn(L)uuut . Denoting by Xt the space spanned byχi t , i ∈ N, if Xt is finite dimensional, then χχχnt has a static representation χχχnt =An fff t , with fff t = N (L)uuut .

Proof. Let r be the dimension of Xt . Stationarity of χχχnt , for all n, impliesthat r is independent of t . Without loss of generality we can assume that fff t =(χ1tχ2t . . .χr t ) is a basis in Xt , for a given t . Again, stationarity of χχχnt impliesthat, for all t , fff t is a basis for Xt and that χi t = ai fff t , for all i , with ai independentof t . Thus χχχnt = An fff t , where An is n ×r with aj on its j th row. Moreover settingN (L) = Br (L), we have fff t = N (L)uuut . n

Model (5)–(6) implies that the entries of N (L) are rational functions of L . Con-versely, assuming that the entries of N (L) are rational functions of L implies thatthe model can be put in the form (5)–(6). This is fairly obvious. If φj (L) is the leastcommon multiple of the denominators of the entries in the j th column of N (L),then N (L) = N1(L)N2(L)−1, where N1(L) is an r × q moving average (MA)and N2(L) is q × q with the polynomials φj (L)−1 on the main diagonal andzero elsewhere. Thus the following assumption is equivalent to assuming (5)and (6).


Assumption 6. The entries of N (L) are rational functions of L .

Note that if Assumption 6 holds for the vector fff t , i.e., for a static representation,then it holds for all static representations.

Our last problem is a specification of N (L) that makes the model suitable foridentification and estimation of the shocks uuut . A standard solution is the assump-tion that N (L) results from inversion of a VAR, i.e.,

fff t − D1 fff t−1 −·· ·− Dm fff t−m = Ruuut , (8)

where R is an r × q matrix, so that N (L) = (I − D1L −·· ·− Dm Lm)−1 R. Thisassumption implies, as shown in Proposition 2 in Section 3.2, that uuut is identifiedup to an orthogonal matrix. However, the VAR specification also implies thatuuut belongs to the space spanned by present and past values of the variables χi t ,i.e., that uuut is fundamental for the χ ’s. This is the issue that will be thoroughlydiscussed in the next section.

3. FUNDAMENTALNESS OF THE STRUCTURAL SHOCKS

3.1. Response Heterogeneity, n Large and Fundamentalness

3.1.1. Fundamentalness in SVAR Analysis. Let us begin by briefly recallingsome basic notions on fundamental representations of stationary stochastic vec-tors. Assume that the n-dimensional stochastic vector μμμt admits a MA representa-tion, i.e., that there exist a q-dimensional white noise vvv t and an n ×q, one-sided,square-summable filter K (L), such that

μμμt = K (L)vvv t . (9)

If vvv t belongs to the space spanned by present and past values of μμμt we say thatrepresentation (9) is fundamental and that vvv t is fundamental for μμμt (the conditiondefining fundamentalness is also referred to as the miniphase assumption; see,e.g., Hannan and Deistler, 1988, p. 25). With no substantial loss of generality wecan suppose that q ≤ n and that vvv t is full rank. Moreover, for our purpose, we cansuppose that the entries of K (L) are rational functions of L and that the rank ofK (z) is maximal, i.e., q, except for a finite number of complex numbers z. Then(see, e.g., Rozanov, 1967, Ch. 1, Sect. 10; Ch. 2, p. 76) we have the followingresult.

PROPOSITION F. Representation (9) is fundamental if and only if the rank ofK (z) is q for all z such that |z| < 1.

Assuming that (9) is fundamental, all fundamental white-noise vectors zzzt arelinear transformations of vvv t , i.e., zzzt = Cvvv t (see Proposition 2 in Section 3.2).Nonfundamental white-noise vectors result from vvv t by means of linear filters thatinvolve the so-called Blaschke matrices (see, e.g., Lippi and Reichlin, 1994).


A fundamental white noise naturally arises with linear prediction. Precisely, theprediction error

wwwt = μμμt −Proj(μμμt |μμμt−1, μμμt−2, . . .)

is white noise and fundamental for μμμt . As a consequence, when estimating an au-toregressive moving average (ARMA) with forecasting purposes, the MA matrixpolynomial is always chosen to be invertible, which implies fundamentalness.

Fundamentalness also plays an important role in the identification of structuralshocks in SVAR analysis. SVAR analysis starts with the projection of a full rankn-dimensional vector μμμt on its past, thus producing an n-dimensional full rankfundamental white noise wwwt . The structural shocks are then obtained as a lineartransformation Awwwt , the matrix A resulting from economic theory statements,which is tantamount to assuming that the structural shocks are fundamental. Fun-damentalness has here the effect that the identification problem is enormouslysimplified. However, as pointed out in the literature mentioned in the Introduction(see also Section 3.1.2), economic theory, in general, does not provide support forfundamentalness, so that all representations that fulfill the same economic state-ments but are nonfundamental are ruled out with no justification.

Our main point is that the situation changes dramatically if structural analysisis conducted assuming that n > q. Precisely, as we see subsequently, fundamen-talness is a nongeneric property for n = q, whereas it is generic for n > q. Thusthe question “why assume fundamentalness?”, which is legitimately asked whenn = q, is replaced by “why should we care about nonfundamentalness?” whenn > q .

An easy and effective illustration can be obtained assuming that q = 1 and thatthe entries of K (L) = (K1(L)K2(L) . . . Kn(L))′ are polynomials whose degreedoes not exceed s, so that K (L) is parameterized in Rn(s+1). In this case, if n =q = 1, nonfundamentalness translates into the condition that at least one root ofK1(z) has modulus smaller than unity. Continuity of the roots of K1(z) impliesthat if nonfundamentalness holds for a point κκκ in the parameter space it holds alsowithin a neighborhood of κκκ . Existence of points in the parameter space for whichnonfundamentalness holds is obvious; thus fundamentalness is nongeneric.

On the other hand, if n > q = 1, by Proposition F, nonfundamentalness impliesthat the polynomials Kj (z) have a common root. As a consequence, their coef-ficients must fulfill n − 1 equality constraints (see, e.g., van der Waerden, 1953,p. 83). Fundamentalness is therefore generic.1

3.1.2. An Example of Structural Nonfundamentalness. The preceding discus-sion has a forceful macroeconomic counterpart. Let us first adapt to ourframework the classic permanent-income consumption model, as used inFernandez-Villaverde, Rubio-Ramirez, Sargent, and Watson (2007) to illustratenonfundamentalness. With minor changes in notation:

ct = ct−1 +σu(1− R−1)ut ,

yt − ct = −ct−1 +σu R−1ut ,


where ct is permanent consumption, yt is labor income, ut is a white-noise pro-cess, and R is a constant gross interest rate. The authors assume that the variableyt − ct , call it st , is observed by the econometrician, whereas ct is not. From thepreceding equations we obtain

st − st−1 = σu R−1(1− RL)ut , (10)

so that, as R > 1, ut is not fundamental for st . Therefore the VAR for st (the bestthe econometrician can do), that is just a univariate autoregression, would producean innovation that is not the structural shock ut . However, if the econometricianobserves ct , or yt , or the value of the consumer’s accumulated assets, then ut

becomes fundamental (Sect. II of Fernandez-Villaverde et al., 2007). Precisely,ut can be recovered using present and past values of st and another variable,whereas present and past values of st alone are not sufficient.

This extremely simple example contains all the elements we need to motivatefundamentalness of the structural shocks uuut for χχχnt .

1. As a rule nonfundamentalness arises when the econometrician’s informationset is smaller than the agent’s (see Hansen and Sargent, 1980, 1991; also thelearning-by-doing example in Lippi and Reichlin, 1993, can be reformulatedin terms of information sets). In the permanent-income model the agent ob-serves permanent income whereas the econometrician does not.

2. However if any additional variable zt = b(L)ut is observed, then, byProposition F, ut is fundamental for the singular vector (st zt )

′, unlessb(R−1)= 0. For example, if zt = α(ut − βut−1), then β = R is sufficientfor fundamentalness of ut for (st zt )

′.3. In our framework, the agent still observes ct and st , and the econometrician

observes

x1t = st + ξ1t ,

i.e., st plus measurement error. However, we also assume that x1t belongsto a large data set xit = χi t + ξi t , which is observed by the econometrician.The common components χi t can be recovered using our large-dimensionalfactor-model techniques. Moreover, assuming for simplicity that q = 1, as inthe permanent-income example, the unique structural shock ut is fundamen-tal for the vector of the common components, unless all the responses bi (L)fulfill the extremely unlikely constraint bi (R−1) = 0.

In general, if the variables χi t are driven by q shocks, a macroeconomic modelthat contains only q variables, suppose they are χj t , j = 1, . . . ,q, cannot en-sure fundamentalness of uuut , the reason being possible superior information of theagents with respect to present and past values of χj t , j = 1, . . . ,q. However, theinformational advantage of the agents disappears if the econometrician observesa large set of additional macroeconomic variables. The generating process of χj t ,j = q +1, . . . ,n, contains parameters that do not belong to the generating process


of the first q and vice versa. Therefore, with all likelihood, their dynamic re-sponses to uuut are sufficiently heterogeneous, with respect to the first q, to preventthe rank reduction that is, by Proposition (F), equivalent to nonfundamentalness.

3.1.3. Fundamentalness in the Dynamic Factor Model. Based on the preced-ing discussion we assume fundamentalness of uuut for χi t , i ∈ N.

PROPOSITION 1. Under Assumptions 1, 2, and 4, fundamentalness of uuut forχi t , i ∈ N, is equivalent to left invertibility of N (L), i.e., to the existence of aq × r filter G(L) such that G(L)N (L) = Iq . Moreover, under Assumptions 1–5,uuut belongs to the space spanned by present and past values of xit , i = 1, . . . ,∞,i.e., is fundamental for xit , i ∈ N.

Proof. If uuut is fundamental for χi t , i ∈ N, then it is fundamental for fff t ; i.e.,there exists a q × r filter G(L) such that uuut = G(L) fff t = G(L)N (L)uuut . As uuut

is a white noise, G(L)N (L) = Iq . Now assume that G(L)N (L) = Iq . Assump-tion 4 implies that A′

n An is full rank for n sufficiently large. Setting Sn(L) =G(L)

(A′

n An)−1

A′n , we have Sn(L)xxxnt = Sn(L)χχχnt + Sn(L)ξξξnt . Now

Sn(L)χχχnt = G(L)(

A′n An)−1

A′n An fff t = G(L) fff t = G(L)N (L)uuut = uuut .

Therefore uuut lies in the space spanned by present and past values of χχχnt . More-over, Sn(L)ξξξnt = G(L)

(A′

n An)−1

A′nξξξ t converges to zero in mean square by

Assumptions 4 and 5. n

The preceding proof also shows that fundamentalness of uuut for χi t , i ∈ N, isequivalent to fundamentalness of uuut for χχχnt for n sufficiently large. In view ofProposition 1, our fundamentalness assumption will be formulated as follows.

Assumption 7 (Fundamentalness). There exists a q × r one-sided filter G(L)such that G(L)N (L) = Iq .

Obviously, if Assumption 7 holds for a particular static representation then itholds for all static representations.

Starting with representation (5)–(6), if no restrictions hold among the coeffi-cients of the matrices Cnj , we have N (L) = (K (L)K (L)L . . . K (L)Ls)′, whichhas left inverse (�(L)0q . . .0q). Thus, as we can obviously expect, no restric-tions implies “maximum heterogeneity” of the responses to the structural shocksand therefore fundamentalness. To see the effect of restrictions consider again theexample with q = 1 and χi t = ci0ut + ci1ut−1 (see Section 2). If the restrictionci1 = aci0 holds we have r = 1 and N (L) = 1 + aL . In this extreme case As-sumption 7, i.e., |a| < 1, is no less arbitrary as the fundamentalness assumptionin VAR analysis. When restrictions hold but r > q, Assumption 7 rules out lowerdimensional subsets of parameter space.

To introduce our last assumption, a VAR specification for fff t , let us considerthe orthogonal projection of fff t on the space spanned by its past values:


fff t = Proj( fff t | fff t−1, fff t−2, . . . , )+wwwt , (11)

where wwwt is the r -dimensional vector of the residuals. Under our assumptions,wwwt has rank q . Moreover, by the same argument used to prove Proposition 2 (seeSection 3.2), Assumption 7 implies that wwwt = Ruuut , where R is a maximum-rankr ×q matrix.

To get some insight into the orthogonal projection (11), consider again repre-sentation (5)–(6) with no restrictions. The static representation of the model hasr = q(s +1) and

fff t = (K (L)K (L)L . . . K (L)Ls)′uuut .

In particular, assuming that h ≤ s, fff t has the AR(1) representation⎛⎜⎜⎜⎜⎜⎝

fff ∗t

fff ∗t−1

...

fff ∗t−s

⎞⎟⎟⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎝

�1 �2 · · · �s−1 �s

Iq 0q · · · 0q 0q

0q Iq · · · 0q 0q

0q 0q · · · Iq 0q

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎝

fff ∗t−1

fff ∗t−2

...

fff ∗t−s−1

⎞⎟⎟⎟⎟⎟⎠+

⎛⎜⎜⎜⎜⎜⎝

Iq

0q

...

0q

⎞⎟⎟⎟⎟⎟⎠uuut ,

where �j = 0q if j > h. If h > s the order of the VAR is higher (but still finite).Joining this observation with the usual approximation argument, a specification

of fff t as in (8), even with m very small, does not seem to cause a dramatic loss ofgenerality. In what follows we will adopt the VAR(1) specification that follows.

Assumption 7′ (Fundamentalness: VAR(1) specification). The r -dimensionalstatic factors fff t admit a VAR(1) representation

fff t = D fff t−1 + Ruuut , (12)

where D is r × r and R is a maximum-rank matrix of dimension r ×q.

Under (12),

χχχnt = Bn(L)uuut = An(I − DL)−1 Ruuut . (13)

Note that assuming (12) (or a higher order autoregressive equation) is independentof the particular static factors we choose. For example, let gggt = G fff t , where G isr ×r and invertible, be another basis in the space spanned by the χi t ’s. If fff t fulfills(12), then gggt = [G DG−1]gggt−1 + [GR]uuut .

A convenient alternative formulation of Assumption 7′ is as follows.

Assumption 7′′. The r -dimensional static factors fff t admit a VAR(1) represen-tation

fff t = D fff t−1 +εεεt , (14)

where D is r × r and εεεt is a white noise of rank q.


3.2. Alternative Fundamental Representations

Our next result shows that if χχχnt = Cn(L)vvv t is a given fundamental representation,then uuut can be obtained from vvv t by means of a static rotation.

PROPOSITION 2. Consider the common components of model (1)

χχχnt = Bn(L)uuut (15)

under Assumptions 1–7. If

χχχnt = Cn(L)vvv t (16)

for any n ∈ N, where the matrices Cn(L) are nested and vvv t is a q-dimensionalfundamental orthonormal white-noise vector, then representation (16) is relatedto representation (15) by

uuut = Hvvv t ,

Bn(L) = Cn(L)H ′,

where H is a q ×q orthogonal matrix, i.e., H H ′ = Iq .

Proof. Projecting uuut entry by entry on the linear space V−t spanned by present

and past values of vht , h = 1, . . . ,q, we get

uuut =∞∑k=0

Hkvvv t−k +rrr t ,

where rrr t is orthogonal to vvv t−k , k ≥ 0. Now consider that V−t and the space

spanned by present and past values of χi t , i ∈N, call it X−t , are identical, because

the entries of χχχ t−k , k ≤ 0, belong to V−t by equation (16), whereas the entries

of vvv t−k , k ≤ 0, belong to X−t by assumption. The same is true for X−

t and thespace spanned by present and past values of uht , i = 1, . . . ,q, call it U−

t , so thatU−

t = V−t . Hence rrr t = 0. Moreover, serial noncorrelation of the vht ’s implies that

∑∞k=1 Hkvvv t−k is the projection of uuut on V−

t−1, which is zero because V−t−1 = U−

t−1.It follows that uuut = H0vvv t . Orthonormality of uuut implies that H0 is orthogonal, i.e.,H0 H ′

0 = I . n

4. IDENTIFICATION AND ESTIMATION

4.1. Variables of Interest, Identification

Proposition 2 has the consequence that structural analysis in large-dimensionalfactor models can be carried out along the same lines as standard SVAR analysis.Precisely, the following procedure is carried out.

Step A. We select the variables of interest, the first m with no loss of generality.Usually m = q.


Step B. We determine a q-dimensional vector vvv t , which is fundamental for χi t ,i ∈ N, and the corresponding representation χχχmt = Cm(L)vvv t .

Step C. We assume that economic theory implies a set of zero and sign restric-tions that uniquely determines the structural impulse-response functionBm(L) ( just identification), i.e., that economic theory identifies a rota-tion H such that Bm(L) = Cm(L)H ′ and uuut = Hvvv t .

Step D. We construct a consistent estimator Bm(L) that is consistent with rate

max((1/

√n), (1/

√T ))

.

Assuming that the variables of interest have been selected, let us concentrateon step B. Denote by Mχ the r ×r diagonal matrix having μ

χj as entry ( j, j) and

by W χ the n × r matrix whose j th column is a unit-modulus eigenvector of �χ0

corresponding to μχj , so that

W χ ′�χ0 = Mχ W χ ′. (17)

Then define

gggt = 1√n

W χ ′χχχnt . (18)

The entries of gggt are the first r (nonnormalized) principal components of χχχnt .Assumption 4(b) implies that for n large enough gggt is a basis for the space Xt ,thus a vector of static factors. A fundamental representation for χχχmt is now easilyobtained as follows.

1. By Assumption 7′′, gggt = Dgggt−1 + εεεt , where, using �gk = E(gggtggg′

t−k) =(1/n)W χ ′�χ

k W χ ,

D = �g1 (�

g0 )−1 = 1

nW χ ′�χ

1 W χ

(Mχ

n

)−1

. (19)

2. Setting �ε = E(εεεtεεε′t ), we have

�ε = �g0 − D�

g0 D′ = Mχ

n− D

Mχ

nD′. (20)

3. Now let μεj , j = 1, . . . ,q, be the j th eigenvalue of �ε , in decreasing order,

M the q ×q diagonal matrix with√

μεj as its ( j, j) entry, Kj a unit-modulus

column eigenvector corresponding to μεj , and K = (K1 K2 . . . Kq). Defining

vvv t =M−1 K ′εεεt and K = KM,

χχχmt = Qm(I − DL)−1Kvvv t =∞∑h=0

Qm DhKvvv t−h = Cm(L)vvv t , (21)

where, using (17) and defining Im = (Im 0m,n−m)′ (the n × m matrix withzero on the last n −m rows and Im on the first m rows),

Qm = E(χχχmtggg′t )[E(gggtggg

′t )]

−1 = √nI ′

m W χ . (22)


Note that gggt , Qm , D,M, K, and vvv t all depend on n. Note also that vvv t is funda-mental for gggt , i.e., for all the χ ’s, not necessarily for χχχmt . In other words, vvv t canbe linearly recovered using contemporaneous and past values of all the χ ’s, notnecessarily the first m of the χ ’s (see Section 5.3 on this point).

The proof of our consistency result will need that the first q eigenvalues ofthe matrix �ε be distinct and asymptotically bounded away from zero (like theeigenvalues of �

χ0 /n; see Assumption 4(b)).

Assumption 7. There exist constants di and di , i = 1, . . . ,q, such that di >di+1, i = 1, . . . ,q −1, and

0 < di < liminfn→∞ με

i ≤ limsupn→∞

μεi < di .

Let us now briefly discuss step C. The assumption of just identification can beformalized as follows.

1. Start with any representation χχχmt = S1(I − S2L)−1S3ssst = S(L)ssst , where ssst

is fundamental for the χ ’s.2. The restrictions implied by economic theory determine a rule, i.e., a function

F , associating an orthogonal q × q matrix with any triple S1, S2, S3, suchthat

Bm(L) = S(L)F(S1, S2, S3)′, uuut = F(S1, S2, S3)ssst .

In particular, setting H = F(Qm, D,K), we have Bm(L) = Cm(L)H ′,uuut = Hvvv t .

4.2. Estimation

Let us start with some definitions and notation.

1. �xk = 1

T ∑Th=k+1 xxxntxxx ′

nt−k ;2. μx

j is the j th eigenvalue of �x0 ;

3. Mx is the r × r diagonal matrix with μxj as its entry ( j, j);

4. W x is the n × r matrix with the corresponding normalized eigenvectors onthe columns.

The main motivation for using the static factors gggt , as defined in (18), is that gggt

can be approximated in probability by the sample principal components of xxxnt :

gggt = 1√n

W x ′xxxt .

However, our consistency proof is not based on this result. Rather, we will directlydeal with Qm , D, and K, which are defined like Qm , D, and K, respectively, with�

χk , Mχ , and W χ replaced by �x

k , Mx , and W x , respectively. Therefore we can


define C(L) = Qm(I − DL)−1K, H = F(Qm, D,K), and the estimated impulse-response function Bm(L) = Cm(L)H ′.

Let us now state our last assumption and the consistency result.

Assumption 8. Denote by γ xk,i j and γ x

k,i j the entries of �xk and �x

k , respectively.There exists a positive real ρ such that

T E[(γ x

k,i j −γ xk,i j )

2]

< ρ,

for k = 0,1 and for all positive integers T, i , and j .

Given i and j , convergence ofAk,i j = T E[(γk,i j −γk,i j )

2]

for T → ∞ can beobtained under mild conditions on the autocovariances and fourth cumulants ofthe x’s.2 As a specification of such conditions would not be useful here, we simplyassume convergence ofAk,i j , which obviously implies that for some positive realnumbers ρk,i j ,

T E[(γk,i j −γk,i j )

2]

< ρk,i j ,

for all T . Assumption 9 requires the additional condition that the real numbersρk,i j have a common upper bound ρ.

PROPOSITION 3. Denote by bk,i j the entries of the kth lag matrix coefficientof Bm(L). Under Assumptions 1–9, for all k ≥ 0, i = 1, . . . ,m, j = 1, . . . ,q,

|bk,i j − bk,i j | = Op

(max

(1√n,

1√T

)). (23)

Proof. See the Appendix. n

In Section 5 we consider a case of partial identification, in which m = q = 3.The restrictions allow identification of the third column of B3(L), call it B3,3(L),i.e., the entries corresponding to the third common shock, but not of the wholeB3(L). Quite obviously, taking as F any member of the infinite set of functionsfulfilling the restrictions, (23) can be applied to B3,3(L).

In conclusion, (23) applies with just or partial identification. Overidentificationis left to further research.

5. AN EMPIRICAL APPLICATION

We illustrate our structural factor model by revisiting an influential work in SVARliterature, namely, the three-dimensional SVAR estimated in King et al. (1991).The variables are U.S. per capita output, investment, and consumption, partialidentification of the permanent shock and corresponding impulse-response func-tions being achieved by imposing long-run neutrality of the remaining shocks onoutput.


Our exercise is based on a panel of macroeconomic series including the threeseries used by King et al. (1991) with the same sampling period. As we see sub-sequently, three common shocks, i.e., q = 3, are consistent with our data set.Moreover, upon estimation of the common components, the variance of the id-iosyncratic components of output and investment accounts for about 15% of theirtotal variance, the fraction falling to 10% for consumption. Thus, using the sameidentification restrictions applied in King et al. (1991) allows a sensible and in-teresting comparison between our impulse-response functions and those found inKing et al. (1991).

5.1. The Data

The data set is quarterly and is based on the FRED II database, Federal Re-serve Bank of St. Louis, and Datastream. The original data of King et al. (1991)are available on Mark Watson’s home page (www.princeton.edu/mwatson/public.html). We collected 89 series, including data from NIPA (National Income andProduct Accounts) tables, price indices, productivity, industrial production in-dices, interest rates, money, financial data, employment, labor costs, shipments,and survey data. A larger n would be desirable, but we were constrained by boththe scarcity of series starting from 1949 (as in King et al., 1991) and the needto balance data of different groups. To use Datastream series we were forced tostart from 1950:1 instead of 1949:1, so that the sampling period is 1950:1–1988:4.Monthly data are taken in quarterly averages. All data have been transformed toreach stationarity according to the ADF(4) test at the 5% level. Finally, the datawere taken in deviation from the mean as required by our formulas and divided bythe standard deviation to make results independent of the units of measurement.A complete description of each series and the related transformations is availableon request.

5.2. The Choice of r and the Number of Common Shocks

As a first step we have to set r and q. Let us begin with r . We computed thesix consistent criteria suggested by Bai and Ng (2002) with r = 1, . . . ,30. Thecriteria I Cp1 and I Cp3 do not work, because they do not reach a minimum forr < 30; I Cp2 has a minimum for r = 12. To compute PCp1, PCp2, and PCp3we estimated σ 2 with r = 15 because with r = 30 none of the criteria reaches aminimum for r < 30. The criterion PCp1 gives r = 15, PCp2 gives r = 14, andPCp3 gives r = 20. Subsequently we report results for r = 12, r = 15, and r = 18,with more detailed statistics for r = 15. With r = 15, the common factors explainon average 79.7% of the total variance.

Regarding the variables of interest, the common factors explain 85.6% of thetotal variance for output, 84.4% for investment, and 89.4% for consumption. TheBai and Ng estimators were criticized for easily overestimating the number ofstatic factors when the idiosyncratic terms are strongly correlated. As a robustness


check we therefore repeated our exercise with r = 9. Result are available uponrequest. The main conclusions do not change.

Regarding q, the criterion proposed by Hallin and Liska (2007), non–log crite-rion I C1, for different choices of the parameters and the penalty functions, pro-duces values of q within the range 2–5. Thus the value q = 3, necessary to carryout the comparison between our results and those of King et al. (1991), does notconflict with available evidence.

5.3. Fundamentalness

We are interested in the impulse-response functions of per capita output, invest-ment, and consumption, i.e., with no loss of generality, in the matrix C3(L)H ′.The question here is that although Cn(L), which is n × 3, is fundamental byassumption for n sufficiently large, the 3 × 3 matrix C3(L) is not necessarilyfundamental.3 In other words, the common shocks can be recovered using con-temporaneous and past values of the n common components, but we do not knowwhether the first three are sufficient.

Figure 1 plots the moduli of the two smallest roots of det(C3(L)) as a functionof r , for r varying over the range 3–30. Note that for r = 3 all the roots must belarger than unity in modulus, because they stem from a three-variate VAR. Thisis in fact the case for r = 3 and r = 4, but for r ≥ 5 the smallest root declines

FIGURE 1. The moduli of the first and the second smallest roots as functions of r .


FIGURE 2. Frequency distribution of the modulus of the smallest root.

and lies always within the unit circle. For r ≥ 22 even the second smallest rootbecomes smaller than unity in modulus.

Figure 2 reports the distribution of the modulus of the smallest root for r = 15,across 1,000 replications for a standard block bootstrap on the x’s; the length ofthe blocks was chosen to be equal to 22 quarters, large enough to retain the cycli-cal information in the series. The mean value is 0.66. The percentage of estimatedvalues larger than one in modulus is 14.5.

Bootstrapping results strongly favor nonfundamentalness of the structuralimpulse-response function C3(L)H ′. This implies that C3(L)H ′ cannot be ob-tained by estimating a VAR for the three-dimensional vector (χ1tχ2tχ3t ). As weargue in Section 5.4, nonfundamentalness of C3(L) explains some important dif-ferences between our structural impulse-response function and that of King et al.(1991).

5.4. Impulse-Response Functions and Variance Decomposition

Coming to the impulse-response functions, as anticipated earlier we impose long-run neutrality of two shocks on per capita output, as in King et al. (1991). This issufficient to reach a partial identification, i.e., to identify the long-run shock andits response functions on the three variables.

Figure 3 shows the response functions of per capita output for r = 12,15,18.The general shape does not change that much with r . The productivity shock has


FIGURE 3. The impulse-response function of the long-run shock on output for r =12,15,18.

positive effects declining with time on the output level. The response functionreach its maximum value after 6–8 quarters with only negligible effects after twoyears. It should be observed that this simple distributed-lag shape is different fromthe one in King et al. (1991), where there is a sharp decline during the second andthe third years, which drives the overall effect back to the impact value.

In Figure 4 we concentrate on the case r = 15. We report the response functionswith 90% confidence bands for output, consumption, and investment, respectively(confidence bands are obtained by means of the block bootstrap technique men-tioned before). The shapes are similar for the three variables, with a positive im-pact effect followed by important, though declining, positive lagged effects.

Table 1 reports the fraction of the forecast-error variance attributed to the per-manent shock for output, consumption, and investment at different horizons. Forease of comparison we report the corresponding numbers obtained with the (re-stricted) VAR model and reported in Table 4 of King et al. (1991).

At horizon 1, our estimates are smaller. The difference is important for con-sumption: only 0.30 according to the factor model as against 0.88 according tothe King et al. (1991) model. But at horizons larger than or equal to 8 quarters ourestimates are greater, the difference being very large for investment: at horizon20 (5 years) the permanent shock explains 46% of its forecast-error variance


FIGURE 4. The plots represent the impulse-response functions of the long-run shock onoutput (top), consumption (middle), and investment (bottom) for r = 15.

according to King et al. (1991) as against 86% with the factor model. Thus a typ-ical puzzle of the VAR literature, the finding that technological and other supplyshocks explain a small fraction of investment variations even in the medium-longrun, seems to find a solution in our factor model.

As the variance of the idiosyncratic components of output, investment, andconsumption does not exceed 15% of their total variance (see Section 5.2), non-fundamentalness of the structural shocks for (χ1t χ2t χ3t ), as opposed to funda-


TABLE 1. Fraction of the forecast-error variance due to the long-run shock

Dynamic factor model King et al. (1991)

Horizon Output Consumption Investment Output Consumption Investment

1 0.37 0.30 0.07 0.45 0.88 0.12(0.20) (0.26) (0.16) (0.28) (0.21) (0.18)

4 0.57 0.77 0.42 0.58 0.89 0.31(0.18) (0.18) (0.17) (0.27) (0.19) (0.23)

8 0.78 0.87 0.72 0.68 0.83 0.40(0.13) (0.13) (0.13) (0.22) (0.18) (0.18)

12 0.86 0.90 0.80 0.73 0.83 0.43(0.09) (0.11) (0.11) (0.19) (0.18) (0.17)

16 0.89 0.91 0.83 0.77 0.85 0.44(0.08) (0.11) (0.10) (0.17) (0.16) (0.16)

20 0.91 0.92 0.86 0.79 0.87 0.46(0.07) (0.12) (0.09) (0.16) (0.15) (0.16)

mentalness of the King et al. (1991) shocks for (x1t x2t x3t ), appears to play amajor role in explaining such different dynamic profiles.

6. CONCLUSIONS

We have argued that dynamic factor models are suitable for structural macroeco-nomic modeling and provide an interesting alternative to structural VARs.

As we have shown, a large panel with a small number of common shocks allowsthe econometrician to recover the structural shocks under a reasonable assumptionon the heterogeneity of the impulse-response functions. Thus the fundamentalnessproblem, which has no solution in the VAR framework, where m shocks must berecovered using present and past values of m variables, becomes tractable whenthe number of variables exceeds the number of shocks.

Our empirical application revisits a SVAR estimated in King et al. (1991) forU.S. output, investment, and consumption. Using a large panel including suchseries, we estimate a factor model with three common shocks and apply the Kinget al. (1991) identification scheme. Two important outcomes are as follows.

1. The three-dimensional impulse-response function corresponding to output,investment, and consumption, implicit in our estimated factor model, is non-fundamental, an important difference with respect to the VAR estimated inKing et al. (1991).

2. Comparing responses of the permanent shock in King et al. (1991) and thefactor model, we find that long-run effects are much more important in the


second. In particular, the long-run response of investment in the factor modelis almost two times the one estimated in King et al. (1991).

NOTES

1. For a general result see Anderson and Deistler (2008). The proof of their Proposition 1 can beadapted to show that for a generic rational K (L), with given maxima for the orders of numerator anddenominator polynomials, if n > q ≥ 1, then rank(K (z)) = q for all complex numbers z (no matterwhether inside, on, or outside the unit circle).

2. Anderson (1971, Sect. 8.3), discusses the univariate case; see, in particular, Theorem 8.3.3, inwhich convergence is obtained under summability of the squared second moments and fourth cumu-lants. Analogous conditions can be obtained in the multivariate case (for formulas linking the secondmoments of estimated covariances to second and fourth cumulants of the x’s, see Hannan, 1970,pp. 209–211).

3. Note that fundamentalness of C3(L) and of C3(L)H ′ are equivalent.

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Working papers 11467.van der Waerden, B.L. (1953). Modern Algebra, vol. I. Frederick Ungar.

APPENDIX: Proof of Proposition 3

If A is a symmetric matrix we denote by μj (A) the j th eigenvalue of A in decreasing

order. Given a matrix B, ‖B‖ denotes the spectral norm of B; thus ‖B‖ =√μ1(B B′),which is the Euclidean norm if B is a row matrix. We will make use of the Weyl inequality(see, e.g., Stewart and Sun, 1990, Cor. 4.10, p. 203): If A and B are two s × s symmetricmatrices, then

|μj (A + B)−μj (A)| ≤√

μ1(B2) = ‖B‖, j = 1, . . . ,s. (A.1)


PROPOSITION P.

(i) ‖Qm − Qm Jr ‖ = Op

(max((1/

√n), (1/

√T )))

,

(ii) ‖Dk − Jr Dk Jr ‖ = Op

(max((1/n), (1/

√T )))

, for all k ≥ 0,

(iii) ‖K− JrKJq‖ = Op

(max((1/n), (1/

√T )))

,

where Jr and Jq , are diagonal matrices, r × r and q × q, respectively, depending on nand T , whose diagonal entries are equal either to 1 or −1.

Roughly speaking, Proposition P states that Qm , Dk , K approximate Qm , Dk , K, re-spectively. To see why we need Jr and Jq let us observe that the entries of gggt and gggt , andalso the columns of K and K , all result from taking eigenvectors and are therefore iden-tified up to a sign (by Assumptions 4(b) and 8, no multiple eigenvalues can occur). Thus,e.g., Qm converges to Qm but only if we choose the right signs.

As a consequence, the raw impulse-response functions, i.e., the rows of Cm(L) as de-fined in (21), and the entries of vvv t are identified up to a sign:

χmt = Cm(L)vvv t =[

Qm Jr

](I −[Jr DJr

]L)−1 [JrKJq

][Jqvvv t

]

= Qm(I − DL)−1Kvvv t = Cm(L)vvv t ,

with Qm = Qm Jr , D = Jr DJr , K= JrKJq , vvv t = Jqvvv t , Cm(L) = Cm(L)Jq .

On the other hand, χχχmt = Cm(L)vvv t is obviously a fundamental representation. There-fore, by assumption, application of the zero and sign restrictions to χχχmt = Cm(L)vvv t and toχχχmt = Cm(L)vvv t gives the same result; i.e., setting H = F(Qm , D,K), we have

Bm(L) = Cm(L)H ′ = Cm(L)H ′; (A.2)

see the discussion in Section 4.1 (step C at the beginning and point 2 at the end).Last, Proposition P implies that

‖H − H‖ = ‖F(Qm , D,K)− F(Qm , D,K)‖ = Op

(max

(1

n,

1√T

)),

this being a standard result under reasonable regularity assumptions for F (the usual identi-fication schemes, with zero first-impact or long-run restrictions, produce functions F withelementary analytic entries). This result, combined with (A.2), implies Proposition 3.

The proof of Proposition P requires some intermediate results.

LEMMA 1. Denoting by Im the n ×m matrix having the identity matrix Im in the firstm rows and zero elsewhere (see Section 4.1),

(i) 1n ‖�x

k −�xk ‖ = Op

(1√T

), k = 0,1.

(ii) 1√n‖I ′

m

(�x

0 −�x0

)‖ = Op

(1√T

)for any (fixed) m.

(iii) 1n ‖�x

k −�χk ‖ = Op

(max((1/n), (1/

√T )))

, k = 0,1.

(iv) 1√n‖I ′

m

(�x

0 −�χ0

)‖ = Op

(max((1/

√n), (1/

√T )))

for any (fixed) m.


Proof. We have

μ1

((�x

k −�xk )(�x

k −�xk )′)

≤ trace((�x

k −�xk )(�x

k −�xk )′)

=n

∑i=1

n

∑j=1

(γ xk,i j −γ x

k,i j )2.

By Assumption 9,

E

[n

∑i=1

n

∑j=1

(γ x

k,i j −γ xk,i j

)2]

<n2ρ

T,

for all positive integers T and k = 0,1. Statement (i) follows using Chebychev’s inequality.Similarly, we have

trace

(I ′

m

(�x

0 −�x0

)2Im

)=

m

∑i=1

n

∑j=1

(γ x0,i j −γ x

0,i j )2 = Op

( n

T

).

Statement (ii) follows. As for (iii), observe that �xk −�

χk = �x

k −�xk +�

ξk by Assumption 3,

so that 1n

∥∥∥�xk −�

χk

∥∥∥ ≤ 1n

∥∥∥�xk −�x

k

∥∥∥+ 1n

∥∥∥�ξk

∥∥∥. The first term on the right-hand side is

Op

(1√T

)by statement (i), whereas the second is bounded by (1/n)μ

ξ1, which is O

(1n

)by Assumption 5. Statement (iv) is obtained in a similar way, using (ii) instead of (i) and

the upper bound (1/√

n)μξ1 instead of (1/n)μ

ξ1. n

LEMMA 2.

(i)(μx

j /n)

−(μ

χj /n)

= Op

(max((1/n), (1/

√T )))

for any j .

(ii) There exists n such that, for all n ≥ n, ( Mχ

n ) is invertible.

(iii) For any n ≥ n and η > 0, there exists τ(η,n) such that, for T ≥ τ(η,n),(

Mx

n

)is invertible with probability larger than 1 − η; moreover, if

(Mx

n

)−1exists for

n = n∗ and T = T ∗, it exists for all n > n∗ and T > T ∗.

(iv)∥∥∥Mχ

n

∥∥∥ and

∥∥∥∥(Mχ

n

)−1∥∥∥∥, which depend on n, are O(1);

∥∥∥ Mx

n

∥∥∥ and

∥∥∥∥( Mx

n

)−1∥∥∥∥,

which depend on n and T , are Op(1).

Proof. Setting A = �χ0 , B = �x

0 − �χ0 and applying (A.1) we get 1

n

∣∣∣μxj −μ

χj

∣∣∣ ≤1n

∥∥∥�x0 −�

χ0

∥∥∥, which is Op

(max((1/n), (1/

√T )))

by Lemma 1(iii). As for (ii), by As-

sumption 4(b) there exists n such that, for n ≥ n, μχr

n > cr > 0, so that det(

Mχ

n

)= 0.

Turning to (iii), setting A = �x0 , B = �x

0 − �x0 and applying Weyl inequality we get

1n

∣∣μxr −μx

r∣∣≤ 1

n

∥∥∥�x0 −�x

0

∥∥∥, which is Op

(1√T

)by Lemma 1(i). Now, μx

r ≥ μχr , because

�ξ0 is positive semidefinite, so that, for n ≥ n, μx

rn > cr > 0. Hence det

(Mx

n

)is bounded

away from zero in probability as T → ∞. The last part of statement (iii) follows from thefact that the rank of the observation matrix xxxT

n , and therefore the rank of �x0 , is nonde-

creasing in n and T . As for (iv), observe that∥∥∥Mχ

n

∥∥∥= μχ1

n and

∥∥∥∥(Mχ

n

)−1∥∥∥∥= n

μχr

, which


are asymptotically bounded by c1 and 1cr

by Assumption 4(b). Boundedness in probability

of∥∥∥ Mx

n

∥∥∥ and

∥∥∥∥( Mx

n

)−1∥∥∥∥ then follow from statement (i). n

LEMMA 3.

(i) ‖Wχ ′W x (Mx/n)− (Mχ/n)Wχ ′W x‖ = Op

(max((1/n), (1/

√T )))

.

(ii) ‖W x ′Wχ Wχ ′W x − Ir ‖ = Op

(max((1/n), (1/

√T )))

.

(iii) There exist diagonal r × r matrices Jr , depending on n and T , whose diagonalentries are equal to either 1 or −1, such that ‖W x ′Wχ − Jr ‖ = Op (max((1/n),

(1/√

T )))

.

Proof. We have∥∥∥Wχ ′W x (Mx/n)− (Mχ/n)Wχ ′W x

∥∥∥=∥∥∥(1/n)Wχ ′(�x0 −�

χ0

)W x∥∥∥≤

1n

∥∥∥�x0 −�

χ0

∥∥∥. Statement (i) then follows from Lemma 1(iii). As for (ii), set

a = W x ′Wχ Wχ ′W x = W x ′Wχ Wχ ′W x Mx

n

(Mx

n

)−1,

b = W x ′Wχ Mχ

n Wχ ′W x(

Mx

n

)−1 = 1n W x ′�χ

0 W x(

Mx

n

)−1,

c = 1n W x ′�x

0 W x(

Mx

n

)−1 = Mx

n

(Mx

n

)−1 = Ir .

We have ‖a − c‖ ≤ ‖a − b‖+‖b − c‖. Both terms are Op

(max((1/n), (1/

√T )))

, the

first by statement (i), the second by Lemma 1(iii). Turning to (iii), let us denote by wxj and

wχj the j th column of W x and Wχ , respectively. By taking a single entry of the matrix on

the left-hand side of statement (i) we get

1

n

(μx

j −μχi

)w

χ ′j wx

i = Op

(max

(1

n,

1√T

)),

i ≤ r , j ≤ r . Now, for j = i , 1n

(μx

j −μχi

)is bounded away from zero in probabil-

ity, because μχi and μ

χj are asymptotically distinct by Assumption 4(b), whereas μx

j

tends to μχj in probability by Lemma 2(i). Hence the off-diagonal terms of W x ′Wχ are

Op

(max((1/n), (1/

√T )))

. Turning to the diagonal terms, let us first observe that wx ′i

Wχ Wχ ′wxi = 1+ Op

(max((1/n), (1/

√T )))

by statement (ii). But

wx ′i Wχ Wχ ′wx

i = (wx ′i w

χi

)2 +r

∑j=1j =i

(wx ′

i wχj

)2 = (wx ′i w

χi

)2 + Op

(max

(1

n,

1√T

)).

Hence(1−|wx ′

i wχi |)(1+|wx ′

i wχi |)=Op

(max((1/n), (1/

√T )))

, so that 1−|wx ′i w

χi | =

Op

(max((1/n), (1/

√T )))

. n

Proof of Proposition P(i). Seta = Qm Jr = √

nI ′m Wχ Jr , where Jr has been defined in Lemma 3(iii),

b = √nI ′

m Wχ Wχ ′W x = √nI ′

m Wχ Wχ ′W x Mx

n

(Mx

n

)−1,


c = √nI ′

m Wχ Mχ

n Wχ ′W x(

Mx

n

)−1 = 1√nI ′

m�χ0 W x

(Mx

n

)−1,

d = 1√nI ′

m �x0 W x

(Mx

n

)−1 = √nI ′

m W x = Qm .

Let us begin by observing that∥∥I ′

m Wχ(

Mχ)1/2 ∥∥ = ‖I ′

m�χ0 Im‖1/2 depends on the en-

tries of the upper left m ×m submatrix of �χ0 and therefore does not depend on n. Denoting

such fixed quantity by ζ , we have

‖√nI ′m Wχ‖ =

∣∣∣∣∣∣∣∣∣∣√nI ′

m Wχ(

Mχ

n

)1/2(Mχ

n

)−1/2∣∣∣∣∣∣∣∣∣∣≤ ζ

∣∣∣∣∣∣∣∣∣∣(

Mχ

n

)−1/2∣∣∣∣∣∣∣∣∣∣ ,

which is O(1) by Lemma 2(iv). Hence ‖√nI ′m Wχ‖ is O(1), so that we can apply Lemma

3(iii) to get ‖a − b‖ = Op

(max((1/n), (1/

√T )))

and Lemma 3(i) to get ‖b − c‖ =Op

(max((1/n), (1/

√T )))

. Finally, Lemma 1(iv) ensures that ‖c − d‖ = Op

(max(

(1/√

n), (1/√

T )))

. n

Proof of Proposition P(ii). We have D = (1/n)W x ′�x1 W x

(Mx

n

)−1(see (19)) and

Jr DJr = (1/n)Jr Wχ ′�χ Wχ(

Mχ

n

)−1 Jr = (1/n)Jr Wχ ′�χ1 Wχ Jr

(Mχ

n

)−1, where

Jr has been defined in Lemma 3(iii). Set

a = D = 1

nW x ′�x

1 W x

(Mx

n

)−1

,

b = 1

nW x ′�χ

1 W x

(Mx

n

)−1

= 1

nW x ′Wχ Wχ ′�χ

1 Wχ Wχ ′W x

(Mx

n

)−1

,

c = 1

nJr Wχ ′�χ

1 Wχ Jr

(Mx

n

)−1

,

d = Jr DJr = 1

nJr Wχ ′�χ

1 Wχ(

Mχ

n

)−1Jr = 1

nJr Wχ ′�χ

1 Wχ Jr

(Mχ

n

)−1.

By Lemma 1(i) ‖a − b‖ is Op

(1√T

); by Lemma 3(iii) ‖b − c‖ is Op

(max((1/n),

(1/√

T )))

; by Lemma 2(i) ‖c − d‖ is Op

(max((1/n), (1/

√T )))

. This proves the

statement for k = 1. Observing that J 2r = Ir , the extension to the case k > 1 is straightfor-

ward. n

LEMMA 4.

(i) ‖�ε − Jr �εJr ‖ = Op

(max((1/n), (1/

√T )))

, where Jr has been defined in

Lemma 3(iii).

(ii) μεj −με

j = Op

(max((1/n), (1/

√T )))

j = 1, . . . ,r .

(iii) ‖M−M‖ = Op

(max((1/n), (1/

√T )))

.

(iv) M−1 exists for n sufficiently large, and its norm is O(1) as n → ∞; moreover,

‖MM−1 − Iq‖ = Op

(max((1/n), (1/

√T )))

.


(v) There exist diagonal q × q matrices Jq , depending on n and T , whose diagonal

entries are either equal to 1 or −1, such that ‖K ′Jr K − Jq‖ = Op

(max((1/n),

(1/√

T )))

.

Proof. We have �ε = (Mx/n)− D(Mx/n)D′ (see (20)) and Jr �εJr = Jr

((Mχ/n)−

D(Mχ/n)D′) Jr = (Mχ/n)− Jr DJr (Mχ/n)Jr D′Jr . Statement (i) then follows from

Lemma 1(i) and Proposition P(ii). As for (ii), notice first that the eigenvalues of Jr �εJr

are identical to those of �ε . Hence setting A = �ε , B = �ε − Jr �εJr and applying (A.1)

we get∣∣∣με

j −μεj

∣∣∣≤ ∥∥∥�ε − Jr �εJr

∥∥∥. Statement (ii) then follows from (i). Turning to (iii),

we have M2 −M2 =(M−M

)(M+M

). As the second factor is asymptotically

bounded away from zero by Assumption 8, the result follows from statement (ii). State-ment (iv) follows from the fact that με

q > dq > 0 for n sufficiently large by Assumption8 and statement (iii). Finally, result (v) is obtained following the lines of Lemma 3, withAssumption 8 ensuring asymptotically distinct eigenvalues instead of Assumption 4(b). n

Proof of Proposition P(iii). Let us denote by N the diagonal matrix having on thediagonal the smallest r − q eigenvalues of �ε and by K⊥ the r × (r − q) matrix havingon the columns the corresponding eigenvectors, so that �ε = KM2 K ′ + K⊥N K ′⊥. As

μεj = 0 for j > q by Lemma 4(ii), the second term is Op

(max((1/n), (1/

√T )))

. Hence

by Lemma 4(i), ‖KM2 K ′ − Jr KM2 K ′Jr ‖ = Op

(max((1/n), (1/

√T )))

, where Jr

has been defined in Lemma 3(iii). Postmultiplying by KM−1, which is O(1) by Lemma4(iv), we get

‖KM2M−1 − Jr KM2 K ′Jr KM−1‖ = Op

(max

(1

n,

1√T

)).

The desired result is obtained by applying Lemmas 4(iv) and (v) and observing thatJqM−1 =M−1Jq , where Jq has been defined in Lemma 4(v). n

25 OPENING THE BLACK BOX: STRUCTURAL FACTOR MODELS …

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